Juhász Róbert
Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jan;85(1 Pt 1):011118. doi: 10.1103/PhysRevE.85.011118. Epub 2012 Jan 11.
The problem of random walk is considered in one dimension in the simultaneous presence of a quenched random force field and long-range connections, the probability of which decays with the distance algebraically as p(l)≃βl(-s). The dynamics are studied mainly by a numerical strong disorder renormalization group method. According to the results, for s>2 the long-range connections are irrelevant, and the mean-square displacement increases as <x(2)(t)>∼(lnt)(2/ψ) with the barrier exponent ψ=1/2, which is known in one-dimensional random environments. For s<2, instead, the quenched disorder is found to be irrelevant, and the dynamical exponent is z=1 like in a homogeneous environment. At the critical point, s=2, the interplay between quenched disorder and long-range connections results in activated scaling, however, with a nontrivial barrier exponent ψ(β), which decays continuously with β but is independent of the form of the quenched disorder. Upper and lower bounds on ψ(β) are established, and numerical estimates are given for various values of β. Besides random walks, accurate numerical estimates of the graph dimension and the resistance exponent are given for various values of β at s=2.
在存在淬火随机力场和长程连接的情况下，在一维中考虑随机游走问题，长程连接的概率随距离以代数形式衰减，即(p(l)≃βl^{(-s)})。主要通过数值强无序重整化群方法研究动力学。根据结果，对于(s>2)，长程连接无关紧要，均方位移随(<x^{2}(t)>∼(lnt)^{(2/ψ)})增加，其中势垒指数(ψ = 1/2)，这在一维随机环境中是已知的。相反，对于(s<2)，发现淬火无序无关紧要，动力学指数(z = 1)，类似于均匀环境。在临界点(s = 2)处，淬火无序和长程连接之间的相互作用导致活化标度，然而，具有非平凡的势垒指数(ψ(β))，它随(β)连续衰减，但与淬火无序的形式无关。建立了(ψ(β))的上下界，并给出了不同(β)值的数值估计。除了随机游走，还给出了(s = 2)时不同(β)值下图形维度和电阻指数的精确数值估计。
